Symmetries in exact Bohrification

TL;DR

This paper explores the structure of commutative subalgebras within noncommutative C*-algebras in quantum mechanics, revealing that these subalgebras encode significant information about the algebra and its symmetries.

Contribution

It provides a detailed proof that the poset of commutative subalgebras determines a C*-algebra as a Jordan algebra and introduces a new Wigner-type theorem for order isomorphisms.

Findings

C(A) determines A as a Jordan algebra for many C*-algebras

Order isomorphisms of C(B(H)) are (anti) unitarily implemented

C(A) relates to the orthomodular poset of projections P(A)

Abstract

The `Bohrification" program in the foundations of quantum mechanics implements Bohr's doctrine of classical concepts through an interplay between commutative and non-commutative operator algebras. Following a brief conceptual and mathematical review of this program, we focus on one half of it, called "exact" Bohrification, where a (typically noncommutative) unital C*-algebra A is studied through its commutative unital C*-subalgebras, organized into a poset C(A). This poset turns out to be a rich invariant of A. To set the stage, we first give a general review of symmetries in elementary quantum mechanics (i.e., on Hilbert space) as well as in algebraic quantum theory, incorporating C(A) as a new kid in town. We then give a detailed proof of a deep result due to Hamhalter (2011), according to which C(A) determines A as a Jordan algebra (at least for a large class of C*-algebras). As a corollary, we prove a new Wigner-type theorem to the effect that order isomorphisms of C(B(H)) are (anti) unitarily implemented. We also show how C(A) is related to the orthomodular poset P(A) of projections in A. These results indicate that C(A) is a serious player in C*-algebras and quantum theory.